Squeezed state

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Figure 1. The vacuum or coherent states are denoted by V, C the n-photon or thermalized states by N, T the asymmetric states by A the squeezed states by S, PS.

The phase change due to a cycle of changes in squeezed states of light... 

This result is a curious and troubling one for the prospect that there can be a classical B held that has real dynamics. This would imply that non-Abelian symmetry is determined by a Lagrangian of the form ( 32 — /i, 2J, where this is automatically zero by duality. However, if this were the case, we would still have non-Abelian symmetry as a nonLagrangian symmetry. This strongly supports the possibility that the electrodynamic vacuum will continue to exhibit non-Abelian symmetries, such as squeezed states, even if we impose E3 = B3 = 0. 

This suggests that electromagnetism may in fact have a deeper non-Abelian structure. In what follows it is assumed that the B(3 field exists. It is likely that the B 3 field exists only as a manifestation of nonlinear optics. This is an aspect of non-Abelian electrodynamics that has been quite under studied. Later, a discussion of squeezed state operators in connection to non-Abelian electrodynamics is mentioned. However, its role in nonlinear optics is an open topic for work. 

The Heisenberg uncertainty relation (9) imposes basic restrictions on the accuracy of the simultaneous measurement of the two quadrature components of the optical held. In the vacuum state the noise is isotropic and the two components have the same level of quantum noise. However, quantum states can be produced in which the isotropy of quantum fluctuations is broken—the uncertainty of one quadrature component, say, Q, can be reduced at the expense of expanding the uncertainty of the conjugate component, P. Such states are called squeezed states [5,6]. They may or may not be the minimum uncertainty states. Thus, for squeezed states... 

To see the nonclassical character of squeezed states better, let us express the variance ((Ag)2) in terms of the P function... 

It is clear from (28) that g(a) is always positive, since p is a positive definite operator. For a coherent state ao), g(a) = (l/ji)exp(— a — ao 2) is a Gaussian in the phase space Re a, Im a which is centered at a0. The section of this function, which is a circle, represents isotropic noise in the coherent state (the same as for the vacuum). The anisotropy introduced by squeezed states means a deformation of the circle into an ellipse or another shape. 

Now, assuming that the two modes are not correlated at time x = 0, it is straightforward to calculate the variances of the quadrature field operators and check, according to the definition (12), whether the field is in a squeezed state. If the initial state of the field is a coherent state of the fundamental mode and a vacuum for the second-harmonic mode, /0) = wa(0)) 0), for which we have... 

Equation (127) describing the evolution of the system is our starting point for further discussion of the second-harmonic generation. If the initial state of the fundamental mode is not a coherent state but has a decomposition into a number states of the form (125) with different bn, equation (127) is still valid when corresponding bn are taken. It is true, for example, for the initially squeezed state of the fundamental mode. 

The marginal phase distributions are illustrated in Fig. 21a, where we have plotted the phase distribution P(Qa) for the signal mode at the evolution time x = 1 and the phase distribution for the ideal squeezed state for the same squeezing parameter. The mean number of photons for the pump mode is equal to 10. It is clear that quantum fluctuations of the pump mode cause broadening of the phase distribution, but the two-peak structure of the distribution with the peaks at ji/2 is obvious. For large squeezing, the phase distribution of the ideal squeezed state becomes the sum of two symmetrically placed delta functions... 

Here, we discuss two kinds of FD squeezed vacuum. We will present explicit forms of these states, which reveal the differences and similarities between them and the conventional IF squeezed vacuum [68] or FD coherent state. We will show that our states are properly normalized in of arbitrary dimension and go over into the conventional squeezed vacuum if the dimension is much greater than the square of the squeeze parameter. Squeezing and squeezed states in FD Hilbert spaces were analyzed, in particular, by Wodkiewicz et al. [31], Figurny et al. [32], Wineland et al. [33], Buzek et al. [16], and Opatrny et al. [20]. An FD analog of the conventional squeezed vacuum was proposed by Miranowicz et al. [34],... 

The pure state (113) is nonmaximally entangled state it reduces to a maximally entangled state for N l. The entangled state is analogous to the pairwise atomic state [22] or the multiatom squeezed state [23], (see also Ref. 24), predicted in the small sample model of two coupled atoms. 

A possibility of generating the nonclassical (in particular, squeezed) states of the electromagnetic field in the cavity with moving walls was pointed out in several studies in [106,114,124,158-161], The dynamical Casimir force has been interpreted as a mechanical signature of the squeezing effect associated with the mirror s motion [123,125] (see also Ref. 162). 

The creation of photons or specific (e.g., squeezed) states of the electromagnetic held due to the motion of some effective mirrors made of the free electrons moving with (ultra)relativistic velocities was studied [206-207]. Another kind of effective moving mirror consisting of the electron-hole plasma generated in semiconductors under the action of powerful laser pulses was also suggested [208,209]. 

Equation (209) shows that the coupled system turns out in a two-mode squeezed state at t > 0. The properties of this state, as well as of any Gaussian state are determined completely by its covariance matrix... 

If p 1, then z2 . The behavior of the distribution function (214) has been shown [189]. Since the argument of the Legendre polynomial is purely imaginary, (>Jin has no oscillations, in contradistinction to the vacuum squeezed state. 

Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum, Electrodynamics, E. A. Hinds... 

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